I have a polytope in $\mathbf{R}^n$ given by the inequality $\boldsymbol{A}\boldsymbol{x}\le \boldsymbol{b}$, where $\boldsymbol{x}\in\mathbf{R}^n$, and $\boldsymbol{A}$ is a rectangular matrix. I was wondering if there exists a definition of generalized barycentric coordinates that can be obtained directly using $\boldsymbol{A}$ and $\boldsymbol{b}$, without the need to transform it to the V-representation (i.e., without the need to obtain the vertices of the polytope). I tried to find them in several papers (like this one), but it seems that most of them require the vertexes of the polytope. There are some types, like the ones used in Eq. 4 of this paper that even though it uses the normal of the facets, it still needs to iterate through all the facets that contain every vertex.
Moreover, if such a definition exists, could it be applied to general $\boldsymbol{A}$ and $\boldsymbol{b}$? Imagine for example the case where $\boldsymbol{A}$ and $\boldsymbol{b}$ define two facets that are parallel to each other, such that one of the is not "active" (in the sense that it does not contain any vertex of the actual polytope)
Thanks!